Can we construct unbounded categories? If we treat $Set$ as a $0$-category, then $1$-category is universal class (the proper class which contains all the sets), $2$-category is the collection which contains all the classes. And so on... And I have three questions:


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*By this way, can we construct infinitely many categories that for each defined category, we always can construct a higher-level category which contains all the object of all the preceding level (such as the preceding level of universal class is $Set$) (i.e.there is no greatest collection containing all categories)?

*Can we extend this construction to $α$-category("$α$" is an arbitrary ordinal number)? If not, why?

*Can someone recommend any book that talks about my question explicitly?
 A: In light of this question, your other question that I just answered makes a lot more sense.


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*(1. & 2.) Yes, we can. Under the mild assumption that there is a proper class of Grothendieck universes, the following holds: Take any set $x$. Then there is a Grothendieck universe $U_{0}$ such that $x \in U_{0}$. We may now form a class sized category $C_{0}$ inside $U_{0}$. (The reason we want to be able to put $x$ in $U_{0}$ is that in typical applications we want to construct $C_{0}$ from $x$ or at least have that $x$ is an object of $C_{0}$.) In particular, taking $x = U_{0}$, there is a Grothendieck universe $U_{1}$ such that $U_{0} \in U_{1}$. Now, in $U_{1}$, $C_{0}$ is a small category and we may use it to construct another class sized category $C_{1}$ in $U_{1}$. This process can be continued set many times and - under slightly stronger assumptions - even 'class many times', but the latter rarely seems to be of interest.

*(3.) Sure. I suggest that you take a look at Shulman's Set theory for category theory. It is written by a non-set-theorist and hence readable without too much knowledge about the set theoretical details. Unfortunately, it contains some mistakes. Two of them are that the author assumes that every model of $\operatorname{ZFC}$ is well-founded and hence isomorphic to a transitive one via the Mostowksi collapse. This isn't true, but no harm in done by just considering well-founded models in this setting. Another mistake (which is related to the first one) is that the author claims the existence of a model of $\operatorname{ZFC}$ implies the existence of a transitive model of $\operatorname{ZFC}$. This is provably false, but again, you may just assume that transitive models exist in our background universe. Other than that he does a nice job at explaining some set-theoretical foundations of category theory in a manner that should be accessible to people who haven't studied axiomatic foundations like $\operatorname{ZFC}$ before.
A: Hint: look into Grothendieck set theory.
